On the affine Birman -Murakami -Wenzl algebra and its cyclotomic quotients

The Birman-Murakami-Wenzl algebra arises from the field of quantum knot invariants as a quotient of the algebra of (n, n)-tangles in a ball, modulo the relations defining the Kauffman link invariant. In this thesis, we consider the affine Birman-Murakami-Wenzl algebra, an algebra modeled after a quotient of (n, n)-tangles in a solid torus, modulo the same relations.;The first quantum invariant of links was the Jones polynomial. The original approach to this invariant, due to Jones, was to exhibit a quotient of the braid group algebra, with a trace having the Markov property. An alternate approach is to use skein theory. Kauffman introduced a fundamentally different polynomial link invariant via skein theory.;This (Kauffman) polynomial can also be obtained by the Jones approach, with an algebra and a trace. A natural candidate for such an algebra is the algebra of (n, n)-tangles, modulo Kauffman's skein relations, denoted KTn. Birman and Wenzl, and independently Murakami, defined an abstract algebra Wn given by generators and relations modeled on KTn, and then showed that the Kauffman polynomial could be recovered from this algebra. Morton and Wassermann showed that Wn ≅ KTn.;The affine analogue of KTn is the algebra KT&d14;n of tangles in the solid torus, modulo Kauffman skein relations. We define an abstract algebra W&d14;n by generators and relations modeled on KT&d14;n and prove the affine analogue of the Morton-Wassermann theorem, namely W&d14;n ≅ KT&d14;n .;An argument involving descending tangles is used to show surjectivity of the natural map W&d14;n → KT&d14;n . The notion of colored connectors is introduced, and it is shown that any set of affine tangles with distinct colored connectors is linearly independent. This sets the stage to show injectivity of the map W&d14;n → KT&d14;n and to describe two different bases.;A cyclotomic affine BMW-algebra denoted Ck,n is a quotient of W&d14;n of finite rank. We give a basis for this cyclotomic algebra and thus determine its rank.